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EXERCISES FOR THE MINI-COURSE.
A. A. Kirillov
July 2000
1. Formula from [B] for Dim s;t E(g):
1
jW j
X
w2W
det(1 + s
w)
det(1 t
w)
=
l
Y
i=1
1 + st e i
1 t e i +1
:
2. Formula for the q-order of an aфne Weil group:
jW aff (R)j q = jW (R)j q
jW ++
aff j q
where W ++
aff is the subset of W aff whose elements transform C +
aff in a aфne cham-
bre inside C+ .
In particular, for R = An the second factor is
Q n
i=1 (1 q i ) 1 .
3. Problem (from [B]). Show that jW ++
aff j q =
Q l
i=1 (1 q e i ) 1 :
Corollary:
jW aff j q = 1
(1 q) l
l
Y
i=1
[e i + 1] q
[e i ] q
:
4. Formula for the (s; t)-dimensions of isotypic components in E(g) (generaliza-
tion of problem 1 to general irreps of g).
5. Structure of some Weyl groups.
a) The group W (E 6 ) is an extension of Z 2 by the simple group of order 25920,
isomorphic to SO+ (Q(E 6 )=3P (E 6 ); B) where the symmetric bilinear form B comes
from the Killing form.
b) The group W (E 7 ) is an extension by Z 2 of the simple group of order 2 9
3 4
5
7
Sp(Q(E 7 )=2P (E 7 ); B) where the alternate bilinear form B comes from the Killing
form.
Note, that Sp(6; F 2 ) ' O(7; F 2 ) for an appropriate quadratic form.
c) The group W + (E 8 ) is an extension by Z 2 of the simple group isomorphic to
SO+ (Q(E 8 )=2P (E 8 ); B) where the alternate bilinear form B comes from the half
of the standard form.
1991 Mathematics Subject Classication. 15A30; 22E60.
Key words and phrases. root systems, semi-simple Lie algebras, invariants, representations of
semi-simple Lie algebras.
Typeset by A M S-T E X
1

2 A. A. KIRILLOV
6. Formula for the dimensions of unirreps of Sp(2n); SO(2n); SO(2n + 1) in
terms of the partition dened by the highest weight:
dim =
Y
(i;j)2
???:
Ref. Jour. of Algebra, 2000.
7. Prove that there is exactly 2 n 1 Coxeter elements in W (An ).
8. Prove that the action of the Borel subgroup B on the tangent space to the
ag manifold F = G=B is equivalent to the coadjoint action of B in b
 :
9. Double Bruhat cells as symplectic leaves of the appropriate Poisson structure
on F .
10  . Comparative statistics of double Bruhat cells and coadjoint orbits.
11. Generating function for the quantity n(w) := dimker(w 1) for W (An ):
X
w2W
q n(w) =
n
Y
k=1
(q + n k) = q
n
+ q 2
n
n 1
X
k=1
1
k
+ : : : + q n
n! :
Hint. Show that n(w) is equal to the numer of cycles in the permutation w.
12. The substitutional exponent and substitutional logarithm.
Let  =
P
k1  k x k
d
dx be a formal vector eld, f(x) = exp  = x(1+
P
k1 a k x k )
the corresponding formal transformation.
Then f k g and fa k g are related by one-to-one transformations called substi-
tutional exponent and substitutional logarithm. The problem is to nd a
manageable expession for these maps.
In particular a 1 () =  1 ; a 2 () =  2 +  2
1 ; a 3 =  3 + 5
2
 1  2 +  3
1 ; : : : .
Hint. If  k are all zero except n , then f(x) = x
n p
1 nnx n
.
References
[B] Bourbaki N., Groupes et algebres de Lie, Hermann, Paris, 1968, 19?? (english edition).
[H] Humphreys J., Introduction to Lie Algebras and Representation Theory, Springer, New
York, 1972, 1980 (second edition).
[K] Kostant B., Lie group representations on polynomial ring, Amer. J. Math. 85 (1963),
327-404.
[OV] Onishchik A. and Vinberg E., A seminar on Lie groups and algebraic groups, Nauka,
Moscow, 1988, 1995 (second edition), pp. 344 (In Russian); English translation: Series in
Soviet Mathematics. Springer-Verlag, 1990, xx+328 pp.
Department of Mathematics, The University of Pennsylvania, Philadelphia, PA
19104-6395 E-mail address: kirillov@math.upenn.edu